Solutions of nonlinear difference equations in the domain of (ζ<inf>n</inf>) -Cesàro matrix in ℓ<inf>t</inf><inf>(</inf><inf>⋅</inf><inf>)</inf> of nonabsolute type, and its pre-quasi ideal

Abstract

We have constructed the sequence space (Ξ (ζ, t)) υ, where ζ= (ζl) is a strictly increasing sequence of positive reals tending to infinity and t= (tl) is a sequence of positive reals with 1 ≤ tl< ∞ , by the domain of (ζl) -Cesàro matrix in the Nakano sequence space ℓ(tl) equipped with the function υ(f)=∑l=0∞(|∑z=0lfzΔζz|ζl)tl for all f= (fz) ∈ Ξ (ζ, t). Some geometric and topological properties of this sequence space, the multiplication mappings defined on it, and the eigenvalues distribution of operator ideal with s-numbers belonging to this sequence space have been investigated. The existence of a fixed point of a Kannan pre-quasi norm contraction mapping on this sequence space and on its pre-quasi operator ideal formed by (Ξ (ζ, t)) υ and s-numbers is presented. Finally, we explain our results by some illustrative examples and applications to the existence of solutions of nonlinear difference equations.